zheng-xu he and o. schramm- hyperbolic and parabolic packings

8/3/2019 Zheng-Xu He and O. Schramm- Hyperbolic and Parabolic Packings

1/27

Discrete Comput Geom 14:123-149 (1995) Discrete & C omputationalGeometry9 1995 Springer-Verlag New Y o r k I n c .

Hyp erbolic and Parabolic Packings*Z h e n g - X u H e 1 a n d O . S c h r a m m 21University of California at San D iego,La Jo l la , CA 92093, US [email protected] Department, The Weizmann Inst i tute,Rehovot 76100, [email protected]. i l

A b s t r a c t . T h e c o n t a c t s g r a p h , o r n e r v e , o f a p a c k i n g , i s a c o m b i n a t o r i a l g r a p ht h a t d e s c r i b e s t h e c o m b i n a t o r i c s o f t h e p a c k in g . L e t G b e t h e 1 - s k e le t o n o f at r i a n g u l a t i o n o f a n o p e n d i s k . G i s s a i d to b e C P p a r a b o l i c ( r e sp . C P h y p e r b o l i c ) i ft h e r e i s a l o c a l l y f i n i te d i s k p a c k i n g P i n t h e p l a n e ( r e s p . t h e u n i t d i s k ) w i thc o n t a c t s g r a p h G . S e v e r a l c r it e r i a f o r d e c i d in g w h e t h e r G i s C P p a r a b o l i c o r C Ph y p e r b o l i c a r e g i v e n , i n c l u d i n g a n e c e s s a r y a n d s u f f i c ie n t c o m b i n a t o r i a l c r i te r i o n , Ac r i t e r i o n in t e r m s o f th e r a n d o m w a l k s ay s t h a t i f t h e r a n d o m w a l k o n G i sr e c u r r e n t , t h e n G i s C P p a r a b o l i c . C o n v e r s e ly , i f G h a s b o u n d e d v a l e n c e a n d t h er a n d o m w a l k o n G i s t r a n s ie n t , t h e n G i s C P h y p e r b o l i c .

W e a l s o g iv e a n ew p r o o f t h a t G i s e i t h e r C P p a r a b o l i c o r C P h y p e r b o l i c , b u tn o t b o t h . T h e n e w p r o o f h as t h e a d v a n t a g e o f b e i n g a p p l i c a b l e t o p a c k i n g s o f m o r eg e n e r a l s h a p e s . A n o t h e r n e w r e s u l t is t h a t i f G i s C P h y p e r b o l i c a n d D i s a n ys i m p l y c o n n e c t e d p r o p e r s u b d o m a i n o f th e p l a n e , t h e n t h e r e i s a d i s k p a c k i n g Pw i t h c o n t a c t s g r a p h G s u c h t h a t P i s c o n t a i n e d a n d l o c a l l y f i n i te i n D .

1 . I n t r o d u c t i o n

W e c o n s i d e r p a c k i n g s o f c o m p a c t c o n n e c t e d s e ts in t h e p l a n e C = ~ 2 o r i n th eR i e m a n n s p h e r e ( ~ = S 2.

G i v e n a n i n d e x e d p a c k i n g P = ( P v : v ~ V ) , i ts c o n t a c t g r a p h , o r n e r v e G = G ( P ) ,i s d e f i n e d a s f o l l ow s . T h e s e t o f v e r t i c e s o f G i s V , t h e i n d e x i n g s e t f o r P , a n d a n

* Bo th au thors acknowledge support by NS F grants. The first author w as also su ppo rted by theA. Sloan Research Fellowship.


2/27

124 Zheng-Xu He and O. Schramme d g e [ v , u ] a p p e a r s i n G p r e c i s e l y w h e n t h e s e t s P v a n d Pu i n t e r s e c t . T h u s Ge n c o d e s s o m e o f th e c o m b i n a t o r i c s o f P . I f al l th e s e t s P v a r e s m o o t h d is k s I i n C ,t h e n i t i s e a s y t o s e e t h a t t h e c o n t a c t s g r a p h i s p l a n a r .T h e c i r c l e - p a c k in g t h e o r e m [ 16 ] s a y s t h a t f o r a n y f in i te p l a n a r g r a p h G t h e r e iss o m e p a c k i n g o f ( g e o m e t r i c ) d is k s in t h e p l a n e w h o s e c o n t a c t s g r a p h i s G . T h isf a n t a s t i c t h e o r e m h a s r e c e i v e d m u c h a t t e n t i o n s i n c e T h u r s t o n c o n j e c t u r e d t h a t t h eR i e m a n n m a p f r o m a s im p l y c o n n e c t e d d o m a i n t o th e u n i t d is k ca n b e a p p r o x i m a t e du s i n g c ir c le p a c k i n g s w i t h p r e s c r i b e d n e r v e s . T h e c o n j e c t u r e w a s l a t e r p r o v e d b yR o d i n a n d S u l l i v a n [2 0]. S o m e p r o o f s o f t h e c i r c l e - p a c k i n g t h e o r e m a p p e a r i n [1 ], [2 ],[28, Chapter 13], [18], [10], [4], [13], [6], [7], [21], [24], and [23].

H e r e , w e a r e c o n c e r n e d w i t h in f i n it e p a c k i n g s . S u p p o s e , f o r e x a m p l e , t h a t G i s ad isk t r iangu la t ion graph; t h a t is , t h e 1 - s k e l e t o n o f a tr i a n g u l a t i o n o f a n o p e nt o p o l o g i c a l d i sk . B y t a k i n g a H a u s d o r f f l im i t o f p a c k i n g s c o r r e s p o n d i n g t o f in i tes u b g r a p h s o f G , a n i n fi n i te p a c k i n g P o f d i s k s in C w h o s e c o n t a c t s g r a p h i s G c a nb e o b t a i n e d . A f e w q u e s t i o n s t h e n n a t u r a l l y a r i se a b o u t t h e p r o p e r t i e s o f P . C a n Pb e b o u n d e d ? C a n P b e l o c a ll y f in i te i n t h e p l a n e ? ( T h i s m e a n s t h a t e v e r y c o m p a c ts u b s e t o f t h e p l a n e i n t e r s e c t s f i n i t el y m a n y o f t h e s e t s i n t h e p a c k i n g . ) T o w h a te x t e n t i s P u n i q u e ?

I t is n o t h a r d t o s e e t h a t ( s ti ll a s s u m i n g G t o b e a d i s k t r i a n g u l a t i o n g r a p h ) t h e r ei s a u n i q u e o p e n t o p o l o g i c a l d i s k D c (~ s u c h t h a t P i s c o n t a i n e d i n D a n d i s l o c a ll yf in i te i n D . T h e b o u n d a r y o f D is j u s t t h e s e t o f a c c u m u l a t i o n p o i n t s o f p . 2 T h i s Di s c a l l e d t h e carrier o f P , a n d is d e n o t e d c a r r ( P ) .I t w a s p r o v e d i n [1 5] t h a t P c a n b e c h o s e n s u c h t h a t c a r r ( P ) is t h e p l a n e o r t h eu n i t d i s k U = { z ~ C : I z l < 1}. B e a r d o n a n d S t e p h e n s o n [3 ] h a v e o b t a i n e d t h isr e s u lt u n d e r t h e a d d i t i o n a l a s s u m p t i o n t h a t G h a s b o u n d e d v a l en c e . 3 T h e r e is as t r o n g u n i q u e n e s s s t a t e m e n t v a l id w h e n c a r r ( P ) = C : a n y o t h e r d i s k p a c k i n g P ' cw i t h n e r v e G i s t h e i m a g e o f P u n d e r a M r b i u s t r a n s f o r m a t i o n [2 2], [ 15 ]. ( T h eM S b i u s g r o u p i s t h e g r o u p g e n e r a t e d b y i n v e r s io n s i n c i r cl e s. I t i s s ix d i m e n s i o n a l . )I n p a r ti c u la r , i t f o ll o w s t h a t t h e r e c a n n o t b e t w o d i sk p a c k i n g s P , P ' w i t h c a r r ( P ) =C , c a r r ( P ' ) = U , a n d G = G ( P ) = G ( P ' ) . I f c a r r ( P ) = U , t h e n t h e r e is a w e a k e rf o r m o f u n i q u e n e s s : a n y d is k p a c k i n g P ' w i t h c a r r ( P ' ) = U t h a t h a s n er v e G is t h ei m a g e o f P u n d e r a M r b i u s t r a n s f o rm a t i o n .

A l l t h i s p a r a l le l s n e a t l y w i t h t h e a n a l y t i c t h e o r y . T h e e x i s t e n ce o f a l o c a ll y f i ni tep a c k i n g i n U o r C i s a d i s c re t e a n a l o g o f th e u n i f o r m i z a t i o n t h e o r e m , w h i c h s a y s t h a ta n y s im p l y c o n n e c t e d n o n c o m p a c t R i e m a n n s u rf a ce is c o n f o rm a l l y e qu i v a le n t t o Co r U . T h e p a r a l l e ls o f t h e u n i q u e n e s s s t a t e m e n t s a r e t h a t a n y c o n f o r m a l m a p f r o mt h e p l a n e i n t o t h e s p h e r e o r f r o m U o n t o U i s a M ~ S biu s t r a n s f o r m a t i o n .

L e t u s s a y t h a t a d i s k t r i a n g u l a t i o n g r a p h G i s C P p a r a b o l i c ( r e s p . C P h y p e rb o l ic )

1 Th e term disk means a geometric disk, a topological disk means a set homeomorphic to acompact disk, and a smooth disk is a topological disk with C t boundary.2 A point z is an accum ulation point of P if every neighborhood of z intersects infinitely man ysets in P.3 Th e valence or degree of a vertex is the number of neighbors it has. "G has bounded valence"means th at th ere is some C < 0o such that every vertex has valence less than C.


3/27

Hyperbolic and Para bolic Packings 125i f t h e r e i s a d i s k p a c k i n g P w i t h c o n t a c t s g r a p h G a n d c a r r i e r c a r r ( P ) = C ( r e s p .c a r r ( P ) = U ) .

W e i n t r o d u c e t h e n o t i o n o f a V E L p a r a b o l i c g r a p h . V E L p a r a b o l i c i ty is ac o m b i n a t o r i a l p r o p e r t y , w h i c h i s d e f i n e d u s in g C a n n o n ' s v e r t e x e x t r e m a l l e n g t h [ 8].T h e p r e c i s e d e f i n it i o n s a p p e a r l a t e r . A g r a p h w h i c h i s n o t V E L p a r a b o l i c i s c a l le dV E L h y p e r b o l i c . W e p r o v e t h a t a d i s k t r i a n g u l a t io n g r a p h i s C P p a r a b o l i c i ff i t i sV E L p a r a b o l i c. T h i s g iv e s a c o m p l e t e c o m b i n a t o r ia l c h a r a c t e r i z a ti o n o f t h e " C Pt y p e " o f a n y d i s k t r i a n g u l a t i o n g r a p h .

U s i n g t h i s e q u i v a le n c e o f C P p a r a b o l i c a n d V E L p a r a b o l ic , w e p r o v e :1 . 1 . T h e o r e m . L e t G b e a d i s k t ri a n g u la t io n g ra p h . I f th e r a n d o m w a l k o n G i sr e c u r r e n t , t h e n G i s C P p a r a b o l i c . C o n v e r s e l y , i f t h e d e g r e e s o f th e v e r t i c e s i n G a r eb o u n d e d a n d t h e r a n d o m w a l k o n G i s t r an s ie n t , t h e n G i s C P h y p e r b o li c .

I t w i ll b e s h o w n t h a t t h e r e a r e C P p a r a b o l i c d i s k t r i a n g u l a t i o n g r a p h s o n w h i c ht h e r a n d o m w a l k is t ra n s i e n t.

W e a l s o g i v e n e w p r o o f s t o t h e a b o v e - q u o t e d r e s u l t s t h a t e v e r y d i s k t r i a n g u l a t i o ng r a p h i s e i t h e r C P p a r a b o l i c , o r C P h y p e r b o l i c , b u t n o t b o t h . T h e r e s u l t s h e r ea c t u a l ly g e n e r a l i z e t h e s e t h e o r e m s , s i n c e t h e p r o o f s a p p l y n o t o n l y t o p a c k i n g s b yg e o m e t r i c d is k s , b u t t o m o r e g e n e r a l s e t s. I n o r d e r t o s t a t e s o m e o f o u r r e s u l ts , w ei n t r o d u c e t h e n o t i o n o f f a t s et s . H e u r i s t i c a l l y , a s e t i s f a t i f i t s a r e a i s r o u g h l yp r o p o r t i o n a l to t h e s q u a r e o f i ts d i a m e t e r , a n d t h is p r o p e r t y a l s o h o l d s l o c a ll y . T h ep r e c i s e d e f i n i t i o n i s :D e f i n i t i o n s [2 6]. T h e o p e n d i s k w i t h c e n t e r x a n d r a d i u s r i s d e n o t e d D ( x , r ) . L e t~" > 0 . A m e a s u r a b l e s e t X c C i s r - fa t i f , f o r e v e r y x ~ X , x ~ o% a n d f o r e v e r yr > 0 s u c h t h a t D ( x , r ) d o e s n o t c o n t a i n X , t h e i n e q u a l i t y

a r e a ( X n D ( x , r ) ) >_ r a r e a ( D ( x , r ) )h o l d s . A p a c k i n g P = ( P ~ : v ~ V ) is f a t i f t h e r e i s s o m e r > 0 s u c h t h a t e a c h P ~ i s~--fat.

F o r e x a m p l e , a n y s m o o t h d i s k is T - fa t f o r s o m e ~" > 0 , a n d i t i s n o t h a r d t o s e et h a t K - q u a s i - d i s k s a r e r ( K ) - f a t [2 6].

W e c a n n o w s t a t e :1 . 2 . T h e o r e m . L e t G = ( V , E ) b e a d i s k t r ia n g u l a ti o n g r a p h , a n d f o r e a c h v E V le tQ ~ c C b e a s m o o t h c o m p a c t t o p o lo g i c a l d i s k . S u p p o s e t h a t th e r e i s s o m e T > 0 s u c ht h a t e a c h Q ~ i s T -fa t. L e t D c C b e a s im p l y c o n n e c t e d d o m a i n , a n d s u p p o s e t h a tD ~ C ( r e s p . D = C ) i f G i s V E L h y p e r b o li c ( r es p . I / E L p a r a b o l ic ) . T h e n t h e re is ap a c k i n g P = ( P o : v ~ V ) i n D , w h i c h i s l o c al ly f i n i te i n D , w h o s e c o n t a c t s g r a p h i s G ,a n d s u c h t h a t Pv i s h o m o t h e t i c t o Q v f o r e a c h v ~ V . 4

C o n v e r s e l y , s u p p o s e t h a t P = ( P v : v ~ V ) i s a f a t p a c k i n g i n C o f s m o o t h d i s k sw h o s e n e r v e is G . T h e n G i s F E L p a r a b o li c i f a n d o n l y i f C - c a r r ( P ) c o n s i s t s o f a s i n g l ep o i n t .

4 "Pv i s homotbet ic to Qv" means tha t there are av > 0 and bv ~ C such that Pv = aoQ v + bo .


4/27

126 Zheng-Xu He and O. SchrammF r o m [ 24 ] w e k n o w t h a t g i v e n a f i ni te p l a n a r g r a p h G * = ( V * , E * ) a n d a s m o o t h

d i s k Q * c C f o r e a c h v ~ V * , t h e r e i s a p a c k i n g P * = ( P * : v ~ V * ) , w i t h G ( P * )= G * a n d P * h o m o t h e t i c t o Q * f o r e a c h v ~ V * . T h i s c o n s t i t u t e s t h e " f i n i t e c a s e "f o r t h e e x is t en c e p a r t i n T h e o r e m 1 .2 . T h e b a s i c in n o v a t i o n h e r e i s t h e c o n t r o l o n eg e t s o n c a r t ( P ) . T h e s i t u a ti o n w h e r e t h e Q v a r e d i sk s , G is V E L h y p e r b o l ic , a n d Dis a n a r b i t ra r y s i m p l y c o n n e c t e d p r o p e r s u b d o m a i n o f C s e e m s i n t e r e s ti n g i n it se lf .

A l t h o u g h t h e e q u i v a l e n c e o f C P p a r a b o l i c it y t o V E L p a r a b o l i c it y g iv e s a c o m -p l e t e c h a r a c t e r i z a t i o n f o r d i s k t r i a n g u l a t i o n g r a p h s , i t i s q u i t e n a t u r a l t o a s k f o ro t h e r c r it e ri a . I t h a s b e e n s h o w n b y B e a r d o n a n d S t e p h e n s o n [5] t h a t i f e v e r y v e r t exi n G h a s d e g r e e g r e a t e r t h a n 7 , t h e n G i s C P h y p e r b o l i c , w h i l e i f e v e r y v e r t e x h a sd e g r e e a t m o s t 6 , G i s C P p a r a b o l i c . W e s h o w t h a t i f f in i te l y m a n y v e r t i c e s i n G h a v ev a l e n c e g r e a t e r t h a n 6 , t h e n G i s C P p a r a b o l i c , w h i l e if t h e l o w e r a v e r a g e v a l e n c e( s e e S e c t i o n 1 0 f o r t h e d e f i n i t i o n ) i n G i s g r e a t e r t h a n 6 , G i s C P h y p e r b o l i c .F r o m R o d i n a n d S u l li v a n' s p r o o f o f t h e l e n g t h - a r e a l e m m a [20 ], i t f o ll o w s t h a t i fYl, ') t2,. 99 i s a s e q u e n c e o f n e s t e d s i m p l e c l o s e d p a t h s i n G a n d E i 1 / 1 7 jl = o :, t h e nG i s n o t C P h y p e r b o l i c . T h i s c a n b e s e e n a s a c r i t e r i o n f o r C P p a r a b o l i c i ty . I nS e c t i o n 9 w e p r e s e n t a c r i t e r i o n o f C P h y p e r b o l i c i t y b a s e d o n a p e r i m e t r i c i n e q u a l i t yi n G . T h e r e w i ll a ls o b e a s o m e w h a t r e s t r i c t e d c o n v e r s e t o t h i s c r i t e ri o n , w h i c h i s i nt h e s p ir it o f R o d i n a n d S u l li v a n 's l e n g t h - a r e a l e m m a .

T h e i n t e r e s t e d r e a d e r m a y w is h t o c o n s u l t S o a r d i ' s p a p e r [2 7], w h i c h s t u d i e sp r o b l e m s r e l a t e d t o t h o s e d i s c u s s e d h e r e .

2 . D i s c r e t e E x t r e m a l L e n g t hI n t h i s s e c ti o n , w e d e f i n e d i s c r e t e e x t r e m a l l e n g t h . L a t e r , a b r i e f d i s c u s s i o n o f t h eh i s t o r y o f t h e s e d e f i n i ti o n s a p p e a r s . W e h a v e c h o s e n t o s t a r t w i t h a n a b s t r a c t n o t i o n ,a n d t h e n s p e c i a l i z e t o m o r e g e o m e t r i c s i t u a t i o n s .Combinatorial Extremal Length. L e t F b e a n o n e m p t y c ol le c ti o n o f n o n e m p t ys u b s e t s o f s o m e s e t X . A ( d i s c re t e ) metric on X i s a fu nc t io n m : X --> [0, o0) . T hearea o f m i s j u st t h e s q u a r e o f th e L 2 n o r m o f m :

a r e a ( m ) = I lm ll 2 = ~ r e ( x ) 2 .x~X

T h e c o l l e c t i o n o f a ll m e t r i c s m o n X w i t h 0 < a r e a ( m ) < oo is d e n o t e d . ~ ' ( X ) .G i v e n a s e t A c X , w e d e f i n e th e length o f A i n t h e m e t r ic m t o b e

L m ( A ) = ~ m ( x ) .x~A

T h i s i s a l s o c a ll e d t h e m - l e n g t h o f A . I f F i s a c o l le c t i o n o f s u b s e t s o f X , w e d e f i n ei ts m - l e n g t h t o b e t h e l e as t m - l e n g t h o f a se t i n F :

L m ( [ ' ) = i n f L m ( A ) .A ~ F


5/27

Hyperb olic an d Parabolic Packings 127F i n a l l y , t h e extremal length o f F i s d e f i n e d a s

{ L m ( r ) 2 }E L ( F ) = s up a r e a ( m ) : m ~ / ( X ) .T h i s i s a n u m b e r i n [0 , o o]. N o t e t h a t t h e r a t i o L m ( F ) Z / a r e a ( m ) d o e s n o t c h a n g e i fw e m u l t i p l y m b y a p o s i ti v e c o n s ta n t . A l s o n o t e t h a t E L ( F ) d o e s n o t d e p e n d o n X ;t h a t i s, t h e v a l u e o f E L ( F ) d o e s n o t c h a n g e i f w e r e p l a c e X w i t h a n y o t h e r s e t t h a tc o n t a i n s e v e r y A ~ F .

T h e v e r i fi c a ti o n o f th e f o ll o w i ng s i m p l e m o n o t o n i c i t y p r o p e r t y o f e x t r e m a l l e n g thi s l e f t t o t h e r e a d e r .2.1. Monotonicity Property. I f each y ~ F contains some Y ' ~ F ' , then E L ( F ) _ >E L ( F 9 .

A t l e a s t w h e n X i s f i n it e , a g e o m e t r i c i n t e r p r e t a t i o n c a n b e g i v e n t o E L ( F ) .C o n s i d e r t h e E u c l i d e a n s p a c e R x o f a ll f u n c t i o n s f : X ---, R . F o r e a c h s u b s e t y o fX , l e t X v ~ R x b e d e f i n e d b y X~(x ) = 1 f o r x ~ y a n d X r ( x ) = 0 o t h e r w i s e . N o wl e t F b e , a s b e f o r e , a c o l l e c t i o n o f s u b s e t s o f X .2 .2 . T h e o r e m ( G e o m e t r i c D e s c r i p t i o n o f E x t r e m a l L e n g t h ). Le t m o be t he po in t o fleas t norm in the convex hul l o f { Xv: 3 ' ~ F}. Then

L m 0 ( F ) 2E L ( F ) [ [ m o J l2.a r e a ( m o )W e d o n o t u s e t hi s t h e o r e m . T h e s i m p l e p r o o f is l e ft to t h e r e a d e r .

Extremal Length in Graphs . I n t h e f o l l o w i n g , G = ( V , E ) i s a l o c a l l y f i n i t e c o n -n e c t e d g r a p h . I t w i ll a l w a y s b e a s i m p l e g r a p h ; t h a t i s, e a c h e d g e h a s t w o d i s t i n c tv e r t ic e s , a n d t h e r e i s a t m o s t o n e e d g e j o i n i n g a n y t w o v e r t ic e s .

A p a t h y i n G i s a f i n i t e o r i n f i n i t e s e q u e n c e ( v 0 , v 1 . . . ) o f v e r t i c e s s u c h t h a t[ U i , U i + l ] E E f o r e v e ry i = 0 , 1 . . . . . T h e e d g e s a n d v e r ti c e s o f y a r e d e n o t e d b yE (y ) = {[v i, v i+ 1 ]: i = 0 , 1 . . . } a n d V ( ~ / ) = { v o , v 1 . . . }, r e s p e c t i v e l y . L i k e w i s e , f o r Fa s e t o f p a t h s i n G , w e s e t V ( F ) = { V ( y ) : y ~ F } a n d E ( F ) = { E ( y ) : y ~ F }. A s e tA c V o f v e r t i c e s is s a i d t o b e connected i f, f o r e v e r y v , w ~ A , t h e r e i s a p a t h y i nG f r o m v t o w w i t h V ( T ) c A . ( W e a l lo w t r iv i a l p a t h s , p a t h s t h a t c o n t a i n o n l y o n ev e r t e x . )

G i v e n s u b s e t s A , B c V , w e l e t F ( A , B ) = F r ( A , B ) d e n o t e t h e s e t o f a l l p a t h si n G w i t h i n i t ia l p o i n t i n A a n d t e r m i n a l p o i n t i n B . W e l e t F v ( A , B ) ( r e s p .F E ( A , B ) ) d e n o t e t h e s e t s o f v e r t i c e s ( r e s p . e d g e s ) o f su c h p a t h s :

F v ( A , B ) = { V ( ~ /) : y ~ F ( A , B ) } ,F E ( A , B ) = { E ( y ) : "y ~ F ( A , B ) } .


6/27

128 Zheng-Xu H e and O. SchrammA fu nc t ion m : V ~ [0, oo) i s ca l l e d a v-metric o n G , a n d a f u n c t i o n m : E ~ [ 0, oo) i sc a l l e d a n e-metric. W h e n m i s a v - m e t r i c ( re s p . a n e - m e t r i c ) w e u s e L m ( 3 ' ) a s as h o r t h a n d f o r L m ( V ( 3 ' )) ( r es p . L m ( E ( 3 ' ) ) ).The ver tex ex t remal l ength V E L a nd edge extremal length E E L be t w e e n A a n d Ba r e d e f i n e d b y

V E L = V E L ~ ( A , B ) = E L ( F v ( A , B ) ) ,E E L = E E L c ( A , B ) = E L ( F E ( A , B ) ) .

T o m a k e t h e d e f in i ti o n o f V E L ( A , B ) m o r e e x pl ic it , w e h a v e

V E L ( A , B ) = s up inf L m ( 3 ' ) 2,~ ~ a r e a ( m )s u p i n f ( E v ~ v ~ , , r e ( v ) ) 2m , E, ~vm( v) 2

H e r e m r u n s o v e r a g ( V ) a n d 3 ' r u n s o v e r F ~ ( A , B ) .T h e s e d e f i n i t io n s g i v e t w o d i s c r e t e a n a l o g s f o r t h e c l as s ic a l n o t i o n o f e x t r e m a l

l e n g t h . ( R e f e r e n c e [ 17 ] i s a g o o d i n t r o d u c t i o n t o c o n t i n u o u s e x t r e m a l l e n g t h . ) A s w ew i ll s e e b e l o w , b o t h a r e u s e f u l . T h e e d g e e x t r e m a l l e n g t h w a s i n t r o d u c e d b y D u f f in ,w h o s h o w e d i n [1 2] t h a t E E L ( A , B ) i s e q u a l t o t h e e l e c t ri c al r e s i s ta n c e b e t w e e n Aa n d B , i f e a c h e d g e i n G i s c o n s i d e r e d t o b e a r e s i s t o r w i t h u n i t r e s is t a n c e . T h ev e r t ex e x t r e m a l l e n g t h w a s i n t r o d u c e d b y C a n n o n [8 ]. C a n n o n ' s m o t i v a t io n w a s t oo b t a i n c r i te r i a f o r d e c i d in g w h e n a g r o u p c a n b e m a d e t o a c t c o n f o rm a l l y o n t h eR i e m a n n s p h e r e C . L a t e r i t w a s d i s c o v e r e d [ 9], [ 25 ] t h a t e x t r e m a l m e t r i c s o f v e r t e xe x t r e m a l l e n g t h ( t h a t is , m e t r i c s r e a l i z in g t h e s u p r e m u m i n t h e d e f i n i t i o n o f t h ee x t r e m a l l e n g t h ) g i v e s q u a r e t il in g s o f r e c t a n g l e s w i t h p r e s c r i b e d c o n t a c t s .

A n i n f i n i te p a t h 3 ' i n G i s transient i f i t c o n t a i n s i n f i n i te l y m a n y d i s t i n c t v e r t i c e s .T h e s e t o f t r a n s i e n t p a t h s i n G t h a t h a v e a n i n i t ia l p o i n t i n A i s d e n o t e d b y F ( A , o0).T h e e d g e a n d v e r t e x e x t r e m a l l e n g t h f r o m A t o oo. a r e d e f i n e d a s

E E L ( A , o o ) = E L ( { E ( 3 " ) : 3 ' ~ F ( A , ~ ) } ) ,V E L ( A , o r = E L ( { V ( 3 ') : 3 ' ~ F ( A , ~ ) } ) .

O f c o u r s e , t hi s m a k e s s e n s e o n l y f o r i n fi n it e G .F o r a v - m e t r i c o r e - m e t r i c m , w e l e t d i n ( A , B ) ( r e s p . d m ( A , ~ ) ) d e n o t e t h e

d i s t a n c e f r o m A t o B ( r e s p . t o oo) i n th e m e t r i c m ; t h a t i s,d . , (A , B ) = L . , ( F ( A , B ) ) = i n f { L . , ( 3 ' ) : 3 ' ~ F ( A , B ) } ,d m ( A , o o ) = L m ( r ( A , o o ) ) = i n f { L m ( 3 ' ): 3 ' ~ F ( A , o o ) } .

A n i n fi n i te g r a p h G i s V E L p a r a b o l i c i f V E L ( { v } , o o ) = oo f o r s o m e v ~ V .O t h e r w i s e , G i s V E L hyperbol ic . Sim i la r ly , G i s E E L p a r a bo l ic i f EEL ({v} , co) = oo fo rs o m e v ~ V , a n d i s EEL hy pe r bo l i c , o t h e r w i s e .


7/27

Hyp erbolic and Parabolic Packings 1292 .3 . R e m a r k . I f V E L ( { v } , oo) = ~ , t h e n a fi n it e a r e a v - m e t r i c m ~ ~ t r( V ) e xi s ts s u c ht h a t d m ( { V } , o o) = o o. T o s e e t h i s , j u s t t a k e r e ( v ) = E ~ l m r ( V ) , w h e r e t h e m e t r ic s m js a t i s f y A ( m j ) < 2 - j a n d L m j ( F ( { v } , o 0 )) = 1 .2 .4 . E x e rc i se . T r y t o d e t e r m i n e V E L ( A , B ) , E E L ( A , B ) , a n d w h e t h e r G i s E E L o rV E L p a r a b o l ic f o r e x a m p l e s o f y o u r c h oi ce .2 .5 . E x e r c i s e . L e t G b e a n i n f in i te c o n n e c t e d g r a p h , a n d le t A c V b e fi n it e a n dn o n e m p t y . S h o w t h a t E E L ( A , oo) = oo i ff G i s E E L p a r a b o l i c , a n d t h a t V E L ( A , oo)= oo i f f G i s V E L p a r a b o l i c .

W h i l e t h e V E L t y p e ( w h e t h e r p a r a b o l i c o r h y p e r b o l i c ) i s m o r e r e l e v a n t t op a c k i ng s , t h e E E L t y p e is c l os e ly r e la t e d t o r a n d o m w a l k s a n d e l e ct ri ci ty . W e d o n o ti n t r o d u c e t h e t e r m i n o l o g y o f e l e c tr ic a l n e t w o r k s h e r e , b u t r e m a r k t h a t a g r a p h G i se l e c t r i c a l l y p a r a b o l i c i f t h e e l e c t r i c r e s i s t a n c e t o i n f i n i ty in t h e g r a p h i s i n f in i t e . ( S e e[11].)

T h e f o l lo w i n g t h e o r e m i s k n o w n .

2 . 6 . T h e o r e m . L e t G = (I ,7 , E ) b e a l o c a ll y f i n i te c o n n e c t e d g r a p h . T h e fo l l o w i n g a r ee q u i v a l e n t :

( 1 ) G i s E E L p a r a b o l ic .( 2 ) G i s e l e c tr i c a ll y p a r a b o l i c .( 3 ) T h e s i m p l e r a n d o m w a l k o n G i s r e c u rr e n t.T h e e q u i v a l e n c e o f (1 ) a n d ( 2 ) i s e s s e n t i a ll y c o n t a i n e d i n [1 2], w h i l e t h e e q u i v a -

l e n c e o f (2 ) a n d ( 3 ) i s g i v e n i n [1 1]. A l s o s e e S e c t i o n 4 o f [2 9] r e g a r d i n g T h e o r e m 2 . 6a n d f u r t h e r e q u i v a l e n t p r o p e r t i e s .

I n S e c t io n 8 w e s e e t h a t V E L a n d E E L p a r a b o l ic i t y a r e c lo s e ly r e l a te d .

3 . T h e P a c k i n g T y p e a n d V e r te x E x t r e m a l L e n g t h3 . 1 . T y p e C h a r a c t e r i z a t i o n T h e o r e m . L e t P = ( P ~ : v ~ V ) b e a f a t p a c k i n g o f( c o m p a c t c o n n e c t e d ) s e ts i n t h e R i e m a n n s p h e re C , a n d l e t G = ( V , E ) b e t h e c o nt a c tsg r ap h o f P . A s s u m e t h a t G i s l o ca ll y f i n it e a n d c o n n e c te d .

( 1 ) I f P i s l o c a l l y f i n i t e i n C - { p } , w h e r e p i s s o m e p o i n t i n C , t h e n G i s V E Lp a r a b o l i c .

( 2 ) C o n v e r s e l y , s u p p o s e t h a t e a c h P o i s a s m o o t h d i s k a n d t h a t G i s a d i s kt r ia n g u l a ti o n g r a p h , w h i c h i s V E L p a r a b o l i c . T h e n P i s l o c a ll y f i n i t e i n C - { p }f o r s o m e p o i n t p E C .

T h e f o l l o w i n g r e s u l t s a b o u t f a t s e t s p r o v e u s e f u l .


8/27

130 Zheng-Xu H e and O. Schramm3 . 2 . O b s e r v a t i o n . L et F b e a z- fat set , z > O. Th en

a r e a ( D ( z , 3 r ) n F ) > r d i a m e t e r ( D ( z , r ) N F ) 2holds for every z ~ C, r > O.P r o o f. L e t x , y ~ D ( z , r ) n F . I t i s c l e a r t h a t D ( x , l Y ~ x l ) c D ( z , 3 r ) . B y t h er - f a tn e s s o f F , w e t h e n h a v e

a r e a ( D ( z , 3 r ) n F ) > a r e a ( D ( x , [ y - x [ ) N F ) > 7 r z [y - x [ 2.T h e o b s e r v a t i o n f o l l o w s . [ ]

T h e f o ll o w i ng le m m a a p p e a r s i n [2 6] .3 . 3 . L e m m a . There i s a pos i t ive func t ion r * : ( 0 , ~ ) - ~ ( 0 , ~ ) such that for every z > 0,fo r every r- fat set A c C, a nd fo r every M Obius transform ation ~: C ---} C the set q~(A)i s z* ( z ) - f a t .

A c e n t r a l i n g r e d i e n t i n t h e p r o o f o f 3.1 i s t h e f o l l o w i n g le m m a , w h i c h w i ll a l s o b eu s e f u l l a t e r .3 . 4 . L e m m a . L e t P = ( P v : v ~ V ) b e a f a t p a c k i n g i n C . L e t G = ( V , E ) d e n o t e t h econtac ts graph o f P , and assu m e that G i s locally f in i te . S uppose that z is anaccum ula t i on po in t o f t he pack ing P t ha t does n o t be long t o U v ~ v Pv . L e t K be acom pac t s e t i n C t ha t does no t con ta in z . F or eve ry A c C l e t V (A ) deno t e the s e t o fvertices v ~ V suc h tha t Pv intersects A . The n

s u p { V E L ( ; ( V ( K ) , V ( W ) ) : W is open and z ~ W I = o~.I n t h e f o l lo w i n g , C ( z , r ) = O D ( z , r ) d e n o t e s t h e c i r c l e w i t h c e n t e r z a n d r a d i u s r .

Proo f . L e t z > 0 b e s u c h t h a t a l l t h e s e t s P o a r e r - f a t . S u p p o s e f i r s t t h a t z # : oo.W e n o w e s t a b l i s h t h a t a n e i g h b o r h o o d o f z i s d i s j o i n t f r o m 13 v ~ v ( x~ P v . L e tR > 0 b e s m a l l e r t h a n t h e d i s t a n c e f r o m z t o K , a n d l e t V ' b e t h e s e t o f v ~ Vs u c h t h a t P o i n t e r s e c t s b o t h c i r c le s C ( z , R ) a n d C ( z , R / 2 ) . S i n c e t h e d i s t a n c e f r o mC ( z , R ) t o C ( z , R / 2 ) i s R / 2 , f o r e a c h v ~ V ( C ( z , R ) ) o V ( C ( z , R / 2 ) ) , w e h a v ed i a m e t e r ( D ( z , R ) n P o ) > R / 2 . T h e r e f o r e , O b s e r v a t i o n 3 .2 sh o w s th a ta r e a ( D ( z , 3 R ) n P v ) > 7 r z R 2 / 4 . I n p a r t i c u l a r , w e s e e t h a t V ( C ( z , R ) ) nV ( C ( z , R / 2 ) ) i s fi n it e . T h i s i m p l i e s t h a t t h e r e i s a n r I ~ ( 0, R / 2 ) s u c h t h a tV ( D ( z , r l ) ) i s d i s j o i n t f r o m V ( C ( z , R ) ) O V ( C ( z , R / 2 ) ) . T h e n i t f o l lo w s t h a t D ( z , r 1 )i s d i s j o i n t f r o m U v ~ v ( r ) P v .

W e d e f i n e i n d u c t i v e l y a s e q u e n c e r 1 > r 2 > - - . o f p o s i t i v e n u m b e r s . T h e f i rs tn u m b e r i n t h i s s e q u e n c e , r I , h a s b e e n d e f i n e d a l r e a d y . S u p p o s e t h a t n > 1 , a n d t h a tr I . . . , r n _ 1 h a v e b e e n d e f i n e d . L e t r n E ( 0 , r ~ _ 1 / 2 ) b e s u f f i ci e n t ly s m a l l s o t h a tV ( C ( z , r ~ ) ) n V ( C ( z , r n _ l / 2 ) ) = O . T h e a r g u m e n t a b o v e s h o w s t h a t s u c h a n r~e x i s t s .


9/27

Hyperb olic and Parabolic Packings 131F o r e a c h n l e t A n b e t h e c l o s e d a n n u l u s b o u n d e d b y C ( z , rn ) a n d C ( z , rJ 2 ) .

D e f i n e a v - m e t r i c m o n G b y s e t t i n gd i a m e t e r ( P v n A n )

r e ( v ) = n= 1 nrn

f o r e a c h v ~ V . B y t h e c o n s t r u c t i o n o f t h e s e q u e n c e r n , a t m o s t o n e t e r m i n t h iss u m i s n o n z e r o . U s i n g t h is a n d O b s e r v a t i o n 3 . 2, w e g e t a n e s t i m a t e f o r t h e a r e a o fm , a s f o l l o w s :

aroa, m , ameter'

d i a m e t e r ( P ~ 0 A n ) 2E En= 1 v e z n r;~ooE E 2 2n= 1 vE V /1 rn

d i a m e t e r ( P ~ r D ( z , r n ) ) 2

o0E E 2 2n r ~n= 1 v~ V

r r - l z - 1 a r e a ( P ~ A D ( z , 3 r n ) )

7 / ' -a r e a ( D ( z , 3 rn ) )r-l E 2 2n = l / 1 r n

=9~ '-1n--~=1 ~-~ < ~ "F i x a p o s i t i v e i n t e g e r N , a n d c o n s i d e r s o m e p a t h y f r o m V ( K ) t o V ( D ( z , r N )) .

F o r e a c h i n t e g e r n ~ [1 , N ) t h e u n i o n U v ~ v P o i s a c o n n e c t e d s e t t h a t i n t e r s e c t st h e t w o c i r c l e s C ( z , rn ) a n d C ( z , rJ 2 ) f o r m i n g t h e b o u n d a r y o f A n . T h e r e f o r e , f o r

1 N - 1s u ch n , E v e y d i a m e t e r ( P ~ ~ A n) >_ rn /2 . T h i s t h e n i m p l i e s L , , ( T ) >_ -~ E n = l l / n ,w h i c h t e n d s t o i n f in i t y a s N ~ oo. S i n c e a r e a ( m ) < o% w e g e t V E L ( V ( K ) ,V ( D ( z , r N ) ) ) --~ ~ , a s N ~ ~ , w h i c h p r o v e s th e l e m m a i n c a s e z 4~ oo.

I t is e a s y to m o d i f y t h e a b o v e a r g u m e n t t o d e a l w i t h t h e c a s e z = ~ . T h e n u m b e r sr l , r 2 , . . , m u s t s a ti sf y i n t h is c a s e D ( O , r 1) ~ U { P~: v ~ V( K) } , Fn+ 1 > 2Fn, a n dV(C(O, 2 r n ) ) ( ") V(C(O, r n + l ) ) = ~ . T h e a n n u l u s A n is d e f i n e d a s t h e a n n u l u s w h o s eb o u n d a r y i s C ( 0 , r n ) to C ( 0 , 2 r n ) . T h e r e s t o f t h e p r o o f r e m a i n s e s s e n t i a l l y t h es a m e . A l t e r n a t i v e l y , u s i n g L e m m a 3 .3 , t h e c a s e z = oo c a n b e r e d u c e d t o t h e c a s ez = 0 . [ ]

W i t h t h i s l e m m a , t h e p r o o f o f t h e f ir s t p a r t o f 3. 1 is e a s y .P r o o f o f 3 . 1 ( 1 ) . S u p p o s e t h a t P i s l o c a l l y f i n i te i n ~ : - { p }. P i c k s o m e v 0 ~ V .A p p l y i n g L e m m a 3 . 4 w i t h K = Pro , z = p , w e s e e k h a t G i s V E L p a r a b o l i c . [ ]


10/27

132 Zheng-Xu H e and O. Schramm4 . S o m e T o p o l o g ic a l L e m m a s

I n t h is s e c t i o n w e g a t h e r a f e w e l e m e n t a r y t o p o l o g i c a l le m m a s , w h ic h w i ll b e n e e d e db e l o w . T h e r e a d e r i s a d v i s e d t o s k ip th e p r o o f s , a n d p e r h a p s r e t u r n t o t h e m l a te r .T h e f o ll o w i n g t w o l e m m a s w i ll e n a b l e u s t o i n fe r t o p o l o g i c a l in f o r m a t i o n o f a

p a c k i n g f r o m t h e c o m b i n a t o r i c s o f t h e c o n t a c t s g r a p h .4 . 1 . N e i g h b o r s S e p a r a t i o n L e m m a . L e t P = - ( P . : v ~ V ) b e a p a c k i n g o f s m o o t hd i s k s i n C , a n d s u p p o s e t h a t t h e c o n ta c t s g r a p h G = ( V , E ) o f P i s a d i s k tr i a n gu la ti o ng r a p h . L e t v o ~ V b e s o m e v e r t e x, a n d l e t N c V - {v o} b e t h e s e t o f n e ig h b o r s o f v o .T h e n t h e r e is a J o r d a n c u r v e Y c U . ~ N P . - P .o t h a t s e p a r a te s P vo f r o mU .~ V - ( lV u { .o } ) P . i n ( ..5

T h e s a m e c o n c lu s io n h o ld s i f G is t h e ( f i n i t e ) 1 - s k el e to n o f a t r ia n g u la ti o n o f a c lo s e dd isk tha t has v o an d a l l i ts ne ighbors as in ter ior ver t ices .P r o o f . N o t e t h a t t h e a s s u m p t i o n s t h a t e a c h P v is s m o o t h i m p l y t h a t t h e i n t e rs e c -t i o n o f a n y t h r e e s e t s in P i s e m p t y . L e t f b e a n e m b e d d i n g o f G i n (~ s u c h t h a t t h ei m a g e o f a n y e d g e [ v 1 v 2] i s c o n t a i n e d i n P v~ u P o2 a n d i s d i s j o i n t f r o m a l l t h e o t h e rs e ts in t h e p a c k i n g . ( T o m a k e a n e x p li c it c o n s t r u c t i o n o f s u c h a n e m b e d d i n g , f o re a c h v ~ V l e t h v : 0 ~ P v b e a h o m e o m o r p h i s m f r o m t h e c l o se d u n i t d is k U c Co n t o P v , a n d f o r e v e r y e d g e [ v I , v 2 ] ~ E l e t P o ~,o 2 b e s o m e p o i n t i n t h e i n t e r s e c t i o nP ~ N P v 2 " W e m a y th e n t a k e f ( [ v l , V 2 ] ) = { h v l ( t h ~ l ( p . . . . 2)): 0 _< t < 1} U{hv2(th~l(po, ,v2)): 0 < t


11/27

Hyperbolic and Parabolic Packings 133Let no, n 2, .. . , n k_ 1 be the neighbor s of v o in circular order aro und v 0, and let y

k - 1be the Jo rda n curve y = U j= 0 f ( [ n j , n j + l] ) , where we take n k = n O T h e curve Y iscon tai ned i n O o e u Po and is disjoint from Poo" We say that two distinct triangles[U1,U2,Ua],[WI,WE,W3] in T n e i g h b o r if they share an edge. If [U1,UE,U3] is atriangle of T that does not conta in v o but neighbors with a triangle c onta inin g v o,

and D o n n lie on opposite sides of the arcay with [Vo, n : , nj+a], then D .. .. .. . 3 0, j, ~+~f( [n j, nj +l ]) . Consequently, D .... 2,~3 is not in the same connecte d compon ent of(~ - y as Po0" If [vl , v2, v 3] and [wl , we, w3] are two nei ghbori ng triangles that donot contai n Vo, then it is clea r that Do~,oz,o3 and D ~ , w 2 , , , 3 are in the same connectedcom pon ent of C - Y. Hen ce it easily follows that for every triangle [vl , v2, v3] thatdoes no t con ta in v 0 the set Dol,O2,o3 is disjoint from the con nec ted co mpo ne nt of(~ - y that contai ns Poo" This implies that 3' separates Poo from U o~v-(uul~o))P~,and the lemma follows since y c O o ~ N Po - Pv0" []4.2. Corollary. L e t G b e a d i s k t r ia n g u l a ti o n g r a p h , a n d l e t P b e a p a c k i n g o f s m o o t hd i s k s i n C w i t h G ( P ) = G . L e t Z b e t h e s e t o f a c c u m u l a t i o n p o i n t s o f P . T h e n t h e re i s ac o n n e c t e d c o m p o n e n t D o f C - Z t h a t c o n t a i n s P , P is l o ca l ly f i n i t e i n D , a n d D i s atopo log ica l d i sk .

This D is called the carrier of P, D = carr(P).The verification of Corollary 4.2 is left to the reader.

4.3. Lemina. L e t P = (Po: v ~ V ) a n d G = (V , E ) b e a s i n L e m m a 4.1, a n d l e tv ~ V, C c V - {u}. S u p p o s e t h a t C is f i n i t e a n d u i s c o n t a i n e d i n a f i n i t e c o m p o n e n t o fG - C . T h e n O o ~ c P o s e p ar a te s P u f r o m t h e s e t o f a c c u m u l a t i o n p o i n t s o f P .P r o o f. L e t V o be the set of vertices that are contained in the same connectedcompo nent of G - C as u is, and let K c C - U o~C Po be a con nec ted set thatintersects Pu- For w ~ V, let N ( w ) c V - {w} denote the neighbors of w in G.From Lemma 4.1 we know that for each w ~ V0 there is a Jordan curve Yw cO o~u(w) P o - P , ~ that s eparates P", from U o~v-(N(,~)u(~})Po. Let Q w denote thecompo nent of C - Yw tha t cont ains P~, and let Q = U o ~ Vo Qo. Suppose thatp ~ K n a Q w , where w ~ V0. Th en p ~ K n Yw- Since K is disjoint from U oEC Poand 3'., ~ U oEN(w) P v , we conclude that p ~ Q~, with w' ~ V0. Thus a Q w N K cQ, for every w E V0. Since V0 is finite, we have o Q c U o E vo a Q o . T h e aboveimplies tha t o Q n K c Q , and because Q is open, o Q n K = 0. Hence Q o K is arelatively open a nd relatively closed subset of K. As Q G K 4= 0 and K is con-nected, we concl ude that K c Q. Because each Qv intersect s finitely many of thesets in the packing P, the lem ma follows. []

4.4. C o n n e c t e d C u t L e m m a . L e t G = ( V , E ) b e t h e 1 - s k e le t o n o f a t ri a n g u la t io n T o fa s im p l y c o n n e c t e d s u r f a c e S . L e t A , B c V b e t w o d i s j o i n t c o n n e c t e d s e t s o f v e rt ic es .S u p p o s e t h a t X c V i n te r s ec t s e v e r y p a t h j o i n i n g A a n d B . T h e n t h e r e i s a c o n n e c t e ds u b s e t o f X t h a t i n t e rs e c ts e v e r y s u c h p a t h .


12/27

134 Zheng-Xu He and O. SchrammT h e l e m m a i s s u r e ly k n o w n , t h o u g h w e h a v e n o t b e e n a b l e t o lo c a t e a r e fe r e n c e .

S i n ce t h e p r o o f o f A l e x a n d e r ' s l e m m a i n [1 9] c a n b e m o d i f i e d t o e s t a b li s h 4 .4 , w e d on o t i n c l u d e a p r o o f h e r e .

5. DualityT h e f o l l o w i n g t h e o r e m a p p e a r s i n [2 5].5 . 1 . D u a l i t y T h e o r e m . L e t G = ( V , E ) be a f i n it e c onne c t e d g raph, and l e t A , B bet w o n o n e m p t y su b s et s o f V . L e t F = F ( A , B ) de no t e t he se t o f a l l pa th s f r om A t o B i nG . L e t F * denote the co l lec tion o f a l l se t s C c V wi th the proper ty tha t each Y E Fintersects C. Then

E L ( F * ) = E L ( F ) - 1A r e l a t e d d u a l i t y t h e o r e m c a n b e f o u n d i n [ 9 ] .W e n e e d o n l y t h e i n e q u a li t y E L ( F * ) < E L ( F ) - 1 , b u t i n t h e f o l lo w i n g sl ig h tl y

m o r e g e n e r a l s e t t i n g , w h e r e t h e g r a p h i s i n f i n i t e .5 . 2 . P r o p o s i t i o n . Le t G = ( V , E ) b e a connec ted graph, poss ib ly inf in ite, l e t A , B c Vbe two none m p ty subs e ts . L e t F be e i the r V ( F ( A , B ) ) o r V ( F ( A , w )). D e n o t e b y F * thecol lec tion o f a l l subse ts 7" c V such tha t T* intersec ts every Y ~ F . Then

E L ( F * ) E L ( F ) _< 1 .P r o o f . I f E L ( F ) = 0 , t h e r e i s n o t h i n g t o p r o v e . S o a s s u m e t h a t m ~ a t ' ( V ) s a ti sf ie sL m ( y ) ~ L > 0 , f o r s o m e L a n d e v e r y T ~ F . F o r v ~ V l e t t h e height o f v b ed e f i n e d a s

h ( v ) = i n f { L . , ( y ) : 3 ' is a p a t h f r o m A t o v } .

F o r t ~ R , l e t V d e n o t e t h e s e t o f v e r t i c e s v E V s u c h t h a t h( v ) - m ( v ) < t


13/27

Hyp erbolic and Parabolic Packings 135T h i s g i v e s

L * 2 L 2a r e a ( m * ) a r e a ( m ) _ < 1 ,

w h i c h p r o v e s t h e p r o p o s i t i o n . [ ]

S u p p o s e n o w t h a t F is s o m e f i ni te n o n e m p t y c o l le c t io n o f f in it e n o n e m p t y s u b s et so f s o m e s e t X . L e t F * d e n o t e t h e c o l l e c t io n o f a ll s u b s e t s o f X t h a t i n t e r s e c t e a c h3 ' ~ F . I t i s n o t d i f fi c u lt t o s e e t h a t E L ( F ) E L ( F * ) > 1 . ( C o n s i d e r t h e g e o m e t r i ci n t e rp r e t a ti o n , T h e o r e m 2 .2 , o f c o m b i n a t o r ia l e x t re m a l l e n g th . ) T h e e x a m p l e F ={ {1 , 2 }, { 2, 3 }, { 3 , 1} } , F* = F , s h o w s t h a t t h e i n e q u a l i t y m a y b e s t r ic t . H e n c e d u a l i t yf a i l s i n t h e p u r e l y c o m b i n a t o r i a l s e t t i n g .

6. CP Hyp erbolic Implies VEL Hyp erbolicP roof o f 3.1 (continued). I t re m a i n s t o p r o v e t h e s e c o n d p a r t o f t h e t h e o r e m . W en o w a d o p t t h e a s s u m p t i o n s o f 3 .1 (2 ). L e t Z b e t h e s e t o f a c c u m u l a t i o n p o i n t s o f P .O u r i m m e d i a t e g o a l is t o v e r i f y t h a t Z is c o n n e c t e d . L e t V1 c V2 c . . . be as e q u e n c e o f f in i t e s u b s e t s o f V s u c h t h a t V = U n V ~ . Fo r e a c h n , l e t ~Qn b e t h e s e to f v e r t i c e s in th e i n f in i te c o n n e c t e d c o m p o n e n t o f G - V ~ , a n d l e t Q n d e n o t e t h ec l o s u re o f U ~ Q , P ~ . C l e a r ly , w e h a v e 0 1 ~ Q 2 D - . . , a n d e a c h se t Q n i s c o m p a c ta n d c o n n e c t e d . N o t e t h a t Z = n n Q n . S i n c e a n e s t e d i n t e r s e c t i o n o f c o m p a c tc o n n e c t e d s e t s is c o n n e c t e d , i t f ol lo w s t h a t Z i s c o n n e c t e d .

L e t u E V b e s o m e v e rt e x . N o r m a l i z i n g w i t h a M S b i u s t r a n s f o r m a t i o n , w ea s s u m e t h a t { z ~ C : [ zl > 1 } u { ~ } i s c o n t a i n e d i n Pu. L e m m a 3 . 3 s h o w s t h a t t h i sd o e s n o t i n v o l v e a n y l o s s o f g e n e r a l it y .

L e t m b e t h e v -m e t r i c o n G d e f in e d b y

m ( v ) = ( ~ ia m ete r(P v) f o ro r v=u.V#U'

L e t ~- > 0 b e s u c h t h a t e a c h P~ is T -f at . S i n c e Po c D ( 0 , 1 ) f o r v 4, u , w e h a v e

a r e a ( m ) = ~ d i a m e t e r ( P v ) 2v~V-{u)_< ~ r - l z - 1 ~ a r e a ( P ~ )vEV-{u}< 7 r - l z - l a r e a ( D ( O , 1 )) < oo.


14/27

136 Zheng-Xu He and O. SchrammL e t C b e a n y f i n it e s u b s e t o f V - {u} s u c h t h a t u i s d i s j o i n t f r o m t h e i n f i n i te

c o m p o n e n t o f G - C . F r o m L e m m a 4 .3 it f o l lo w s t h a t t h e u n i o n U ~ ~ c P ~ s e p a -r a t e s P ~ f r o m Z . T h i s c l e a r l y i m p l i e s t h a t

m ( v ) > di am et er ( Z) . ( 6 . 1 )v~ C

S i n c e G i s V E L p a r a b o l i c a n d A ( m ) < 0% P r o p o s i t i o n 5 . 2 i m p l i e s t h a tinf Y'~ m ( v ) = O,S v~S

w h e r e t h e i n f i r n u m r u n s o v e r a l l s e t s S c V s u c h t h a t u i s n o t i n a n i n f i n i t ec o m p o n e n t o f G - S . H o w e v e r , e v e r y s u c h S c o n t a i n s a f i n it e C c S s u c h t h a t u isn o t in th e ' i n f in i t e c o m p o n e n t o f G - C ( f o r e x a m p l e , t h e n e i g h b o r s o f t h e c o m p o -n e n t o f G - S c o n t a in i n g u ) . T h e r e f o r e , (6 .1 ) s h o w s t h a t d i a m e t e r ( Z ) = 0 , a sr e q u i r e d . [ ]

W e e a s i l y g e t t h e f o l l o w i n g g e n e r a l i z a t i o n o f 3 .1 (2 ).6 . 1 . T h e o r e m . L e t r > 0 , / e t P = (P v : v ~ V ) be a pack ing o f r - fa t se t s in theR ie m ann s phe r e C , a nd l e t G = ( V , E ) be t he c on tac ts g r aph o f P . As s u m e tha t G i sconnec ted , locally f in i te , and VE L parabol ic . A lso suppose tha t for each u ~ V there i sa Jordan curve 3 ' c U v ~ N( , ) P~ - Pu tha t separates Pu f ro m U v ~ v - ( N ( u ) u ( u ) ) P~ inC . H e r e N ( u ) de no t e s t he s e t o f ne ighbors o f u . T he n t he s e t o f ac c um u la t i on po in t s o f Phas zero length . I f G has o ne e nd thi s se t cons is ts o f a s ing le po in t .

W e r e c a l l t h a t a g r a p h G = ( V , E ) h a s o n e e n d i f f G - K h a g o n e i n f i n i tec o m p o n e n t f o r e v e ry f i ni te K c V .

F o r e x a m p l e , if P i s a l o c a ll y f i n it e t il i n g o f a d o m a i n ~ c C b y c o m p a c t s q u a r e s ,a n d i f t h e c o n t a c t s g r a p h i s c o n n e c t e d , t h e n a l ~ h a s z e r o l e n g t h . I n t h i s c a s e t h ec o n t a c t s g r a p h d o e s n o t h a v e t o b e p l a n a r , s in c e f o u r s q u a r e s m a y m e e t a t a p oi n t.

Proof . T h e p r o o f i s e s s e n t i a l l y t h e s a m e a s f o r 3 .1 (2 ). N o t e t h a t t h e a s s u m p t i o n st h a t t h e s e t s Po a r e s m o o t h a n d t h a t G is a d i s k t r i a n g u la t i o n g r a p h w e r e u s e d o n l yi n th e p r o o f o f L e m m a 4 .1 . S i n c e w e a r e a s s u m i n g t h e c o n c lu s i o n o f th is l e m m ah e r e , t h e s e a s s u m p t i o n s a r e n o t n e e d e d . I f G h a s o n e e n d , t h e n i t i s e a s y t o s e e t h att h e s e t o f a c c u m u l a t i o n p o i n t s o f P i s c o n n e c t e d . T h e p r o o f o f 3 .1 (2 ) s h o w s th a t i no u r p r e s e n t c a s e f o r e v e r y e > 0 th e s e t o f a c c u m u l a t i o n p o i n t s o f P i s c o v e r e d b y af i n it e c o l l e c t i o n o f s e t s s u c h t h a t t h e s u m o f t h e i r d i a m e t e r s i s l e s s t h a n e . T h i sc l e a rl y i m p l i e s T h e o r e m 6 .1 . [ ]

7. Un i formizat ions of Packings7 .1 . Uniformizat ion Theorem. Le t G = (V , E ) be a d i sk tr iangulat ion graph, fo re ac h v ~ V l e t Q , c C be a s m oo th d i s k , and l e t D c C be a s im p ly c onne c t e d dom a in .


15/27

Hyp erbolic and Parabolic Packings 137A s s u m e t h a t th e r e i s a z > 0 s u c h t h a t Q o i s z - fa t f o r e a c h v ~ V . A l s o s u p p o s e t h a tD ~ C ( r e s p . D = C ) i f G i s V E L h y p e r b o l ic ( re s p . p a r a b o l i c ). T h e n t h e r e i s a p a c k i n gP = ( Po : v ~ V ) w i th c a r r ( P ) = D w h o s e c o n t a c t s g r a p h i s G , a n d s u c h t h a t P v i sh o m o t h e t i c t o Q v f o r e a ch v E V .

W e n o t e t h a t t h e c o n t i n u o u s a n a l o g u e o f t h is t h e o r e m a p p e a r s i n [26 ]. T h e p r o o fi s a l so s imi la r .P r o o f . L e t T b e t h e t r i a n g u l a t i o n o f a d i s k t h a t h a s G a s i ts 1 - s k e l e to n . L e tT 1 c T 2 c T 3 c . .- b e a n e x h a u s t i o n o f T . B y t h is w e m e a n t h a t T = U j T j , a n de a c h T j i s a f in i t e t r i a n g u l a t i o n o f a d i s k ( w i t h b o u n d a r y ) . I t is e a s y t o s e e t h a ts u c h a n e x h a u s t i o n e x is ts . W e a l s o re q u i r e , w i t h o u t l o s s o f g e n e ra l i ty , t h a t T 1 h a ss o m e i n t e r i o r v e r te x , s a y v 0 . F o r e a c h j = 1 , 2 , . . . , l e t G j = ( V , E j ) d e n o t e t h e1 - s k e l e t o n o f T j.

S u p p o s e , w i t h o u t l o s s o f g e n e ra l i ty , t h a t 0 ~ D a n d 0 is i n t h e i n t e r i o r o f Q v o " L e tD j b e a s e q u e n c e o f s m o o t h J o r d a n d o m a i n s 6 in C s u c h t h a t 0 ~ D 1 C D E c . . -a n d D = U j D j . F r o m t h e p a c k i n g t h e o r e m s o f [ 24 ] w e k n o w t h a t f o r e a c hj = 1 , 2 . . . . t h e r e is a p a c k i n g P J = ( P J : v ~ V j ) i n t h e c l o s u r e o f D j , s u c h t h a te a c h P J is h o m o t h e t i c t o Q ~ , t h e se ts P / a r e t a n g e n t t o c ~ DJ w h e n v i s a b o u n d a r yv e r t e x o f T j , a n d P J0 h a s t h e f o r m t j Q v o f o r s o m e t j > 0 . L e t { j ( k ) } b e as u b s e q u e n c e o f { 1, 2 . . . . } s u c h t h a t t h e H a u s d o r f f l i m i t

1/5~ = l im p fk ) (7 .1 )k - . = d i a m e t e r ( PJ (k )~~, Vo )e x is ts f o r e v e r y v ~ V . T h e H a u s d o r f f l i m i t i s t a k e n i n C ; t h a t i s , a p r i o r i w e m u s ta ll o w f o r t h e p o s s ib i li ty t h a t oo i s c o n t a i n e d i n s o m e / 5 v .

W e s h o w n o w t h a t t h e s e ts /5v d o n o t d e g e n e r a t e t o s in g le p o i n t s a n d d o n o tc o n t a i n o o. T h e s e t / 5 o c e r t a i n l y i s O K , s i n c e i t c o n t a i n s 0 , h a s d i a m e t e r 1 , a n d i sh o m o t h e t i c t o Q v o , b y c o n s t r u c t i o n . L e t u b e a n y n e i g h b o r o f v 0 . S i n ce / 5 , i s aH a u s d o r f f l im i t o f s et s h o m o t h e t i c t o Q u , w h i c h is s m o o t h , / 5 u is e i t h e r h o m o t h e t i cto Q u , o r i s a s i n g l e p o i n t , o r a h a l f - p l a n e , o r / 5 = ~ . T h e l a s t c a s e i s c l e a r l yi m p o s s i b l e , s i n c e t h e i n t e r i o r o f / 5, d o e s n o t i n t e r s e c t / 5~ 0 . I t is a l s o c l e a r t h a t P ,i n t e r s e c t s /5~ 0 b u t d o e s n o t i n t e r s e c t i t s i n t e ri o r .

L e t u l , u 2 . . . . u , b e t h e n e i g h b o r s o f v 0 , i n c i r c u l a r o r d e r . F o r e v e r y j s u c h t h a tv o a n d a l l it s n e i g h b o r s a r e i n t h e i n t e r i o r o f T j , t h e s e t P j , U - . . U P~ , c o n t a i n s aJ o r d a n c u r v e t h a t s e p a r a t e s P jo f r o m oo. ( T h i s f o l lo w s f r o m L e m m a 4 . 1 .) T h e r e f o r e ,f o r a t l e a s t t w o n e i g h b o r s u o f v 0 t h e s e t s /5 , c o n t a i n m o r e t h a n a s in g l e p o i n t .Su p p o s e , f o r e x a m p l e , t h a t / 5- 1 i s a s in g l e p o i n t p , a n d t h a t / s u , i s n o t a s i n g l e p o i n t .L e t m b e t h e l a r g e s t n u m b e r i n { 1, 2 . . . . . n } s u c h t h a t / 5 ,. = { p} f o r e a c h r < m i n{ 1, 2 . . . . n } . S i n c e a t l e a s t t w o / 5 ,, d o n o t d e g e n e r a t e t o p o i n t s , m < n . I t is c l e a rt h a t e a c h 1 5 i n t e r s e c ts /~ u, a n d t h a t / 5- 1 i n t e r s e c ts P , . T h e r e f o r e , t h e t h r e es m o o t h s e ts P o 0 ' / 5 , / 5 , , c o n t a i n t h e p o i n t p . T h is i m p l ie s th a t t h e i n te r i or s o f t w o

6 A smooth Jordan domain is the interior o f a smooth disk.


16/27

138 Zheng-Xu H e and O. Schrammo f th e s e s e ts m u s t i n t e rs e c t , w h i c h is c l e a rl y im p o s s i b l e . T h u s w e c o n c l u d e t h a t n o n eo f t h e s e t s / ~ , c o n s i s t s o f a s i n g le p o i n t , a n d t h a t t h e r a t i o s d i a m e t e r ( P ~ o ) /d i a m e t e r ( P ~ i ) a r e b o u n d e d f r o m a b o v e . ( T h e r e a d e r m a y w i sh t o c o m p a r e t h e a b o v ea r g u m e n t w i th t h e R i n g L e m m a o f [2 0].)

I s i t p o s s i b l e t h a t / 5 i s a h a l f - p l a n e ? T o s e e t h a t i t i s n o t , c o n s i d e r a H a u s d o r f fl i m i t o f t h e p a c k i n g s ( h j ( P ~ ) : v E v J ) , w h e r e h j is t h e h o m o t h e t y th a t t a k e s P ~ t oQ , I " T h e s a m e a r g u m e n t a s a b o v e , b u t w i t h t h e r o le s o f u 1 a n d v 0 s w i t ch e d , s h o w st h e n t h a t t h e r a t i o s d i a m e t e r ( P ~ , ) / d i a m e t e r ( P ~ o ) a r e b o u n d e d f r o m a b o v e . S i m i la rl y ,f o r e v e r y e d g e [ u , w ] t h e r a t i o d i a m e t e r ( P J u ) / d i a m e t e r ( P ~ ) i s b o u n d e d i n d e p e n d e n t l yo f j . S i n c e G i s c o n n e c t e d , t h is a l s o h o l d s w h e n u , w ~ V a r e n o t n e i g h b o r s .T h e r e f o r e , e a c h s e t / 5 i s n o t a h a l f -p l a n e , n o r a p o i n t, a n d t h u s is h o m o t h e t i c t o Q ~ .

I f G i s V E L p a r a b o l i c , t h e n p a r t ( 2 ) o f 3.1 i m p l i e s t h a t /5 is lo c a l ly fi n it e i n(~ - { p} f o r s o m e p ~ C . I t i s e a s y t o s e e t h a t p = ~ , a n d t h u s / 5 i s l o c a l ly f i ni t e i nC . T h i s c o m p l e t e s t h e p r o o f in th e c a s e t h a t G i s V E L p a r a b o l i c .

N o w s u p p o s e t h a t G i s V E L h y p e r b o li c . T h e s e t P J0 is c o n t a i n e d i n D ~ C a n dh a s t h e f o r m t jQ~o, t j > 0 . S i n c e 0 is a n i n t e r i o r p o i n t o f Q o o , t h i s i m p l i e s t h a t t h es e q u e n c e t j i s b o u n d e d f r o m a b o v e , a n d h e n c e d i a m e t e r (P ~ 0 ) i s b o u n d e d f r o ma b o v e . B y p a s s i n g t o a s u b s e q u e n c e o f j ( k ) , i f n e c e s s a r y , a s s u m e t h a t t =l i m k . = d i a m e t e r ( P J 0 ) ~ [0 , oo) e x is ts . W e h a v e e s t a b l i s h e d a b o v e t h a t f o r a n y v , wV t h e r a t i o s d i a m e t e r ( P ~ ) / d i a m e t e r ( P ~ ) r e m a i n b o u n d e d a s j ~ oo. C o n s i d e r t h eH a u s d o r f f l im i ts

P ~ = l i m P ~ J ( k ) . ( 7 . 2 )k-- . oo

I f t = 0 , t h e n , b e c a u s e G i s c o n n e c t e d , i t f o l lo w s t h a t P ~ = { 0} f o r e a c h v , a n d i np a r t i c u l a r t h e l i m i ts ( 7 .2 ) e x is t. I f t > 0 , t h e n c o m p a r i n g w i t h (7 .1 ), w e c o n c l u d ea g a i n t h a t t h e s e l i m i t s e x is t, a n d t h a t e a c h P ~ i s h o m o t h e t i c t o Q ~ .

W e n o w p r o v e t h a t e a c h P o is c o n t a i n e d i n D . C o n s i d e r s o m e v e r t e x v ~ V , a n dl e t N ( v ) d e n o t e t h e n e i g h b o r s o f v . B y L e m m a 4 . 1, f o r e a c h j s u ff ic i en t ly l a rg e ( sot h a t N ( v ) i s c o n t a i n e d i n th e i n t e r i o r o f T j ) t h e r e i s a J o r d a n c u r v e i n U , ~ ~ v(o) P J- P [ t h a t s e p a r a t e s P [ f r o m a D j. A s s u m i n g t h a t t > 0 , s i n c e f o r a n y f ix e d u t h es e t s P ~ v a r y w i t h in a c o m p a c t co l l e ct i o n o f h o m o t h e t i e s o f Q , , t h e a b o v e i m p l i est h a t t h e d i s t a n c e f r o m P [ t o O D is b o u n d e d f r o m b e l o w i n d e p e n d e n t l y o f j.T h e r e f o r e , P o c D . T h e s a m e c o n c l u s i o n is t r u e , o f c o u r s e , i f t = 0 , b e c a u s e t h e nP ~ = { 0} . S o w e h a v e e s t a b l i s h e d t h a t t h e p a c k i n g P i s c o n t a i n e d i n D .

C l e a r l y , t h e i n t e r i o r s o f t h e s e t s P ~ a r e d i sj o in t , a n d P o n P , ~ O w h e n e v e r[ v , w ] ~ E . T h e r e f o r e , t h e p r o o f w i ll b e c o m p l e t e o n c e w e s h o w t h a t t h e p a c k i n gP = ( P v : v ~ V ) is l o c a l l y f i n i t e in D . ( T h i s w i l l a l s o r u l e o u t t h e p o s s i b i l i t y t = 0 ,P o = { 0} .) T h a t i s a c t u a l l y t h e m o s t s i g n i f i ca n t p a r t o f t h e p r o o f . I t t u r n s o u t t h a t t h ep a c k i n g / 6 i s u s e f u l to p r o v i n g t h is p r o p e r t y o f P .

L e t F b e s o m e c o m p a c t c o n n e c t e d s u b s e t o f D t h a t c o n ta i n s 0. W e p r o v e t h a t Fi n t e r s e c t s f i n i t el y m a n y s e ts in P , a n d t h i s s h o w s t h a t P i s l o c a ll y f i ni t e i n D . L e t F 'b e a n y c o m p a c t c o n n e c t e d s u b s e t o f D t h a t c o n t a in s F i n i ts i n te r io r . L e t z b e s o m ea c c u m u l a t i o n p o i n t o f /5 . F r o m L e m m a 4 .3 w e k n o w t h a t /5 is d i sj o in t f r o m i tsa c c u m u l a t i o n p o i n ts . B y T h e o r e m 3 .1 , z i s n o t t h e o n l y a c c u m u l a t i o n p o i n t o f /5 .


17/27

Hype rbolic and Parabolic Packings 139T h e r e f o r e , t h e r e i s a c o m p a c t c o n n e c t e d s e t K t h a t i n te r s e c ts / 5 0 , c o n t a i n s a na c c u m u l a t i o n p o i n t o f / 5 , a n d i s d i s j o in t f r o m z . I n t h e f o ll o w i n g , f o r a s e t X c ~ , l e tI V ( X ) d e n o t e t h e s e t o f v ~ V s u c h t h a t /5~ i n t e r s e c ts X . S i n ce K i s c o n n e c t e d a n dc o n t a i n s a n a c c u m u l a t i o n p o i n t o f / 5, i t i s c l e a r t h a t e a c h c o m p o n e n t o f I V ( K ) i si n fi n it e . (T h i s f o l l o w s f r o m L e m m a 4 . 3 .)

W e l e t 6 b e a s m a l l p o s it iv e n u m b e r w h o s e v a l u e is d e t e r m i n e d b e l o w . B yL e m m a 3 .4 , t h e r e is s o m e o p e n s e t W = W ( z , K , E ) c o n t a i n i n g z s u c h t h a t

1V E L c ( I V ( K ) , I V (W ) ) > - .E

W i t h o u t l o s s o f g e n e r a li ty , w e a s s u m e t h a t W i s c o n n e c t e d . T h e n e v e r y c o m p o n e n to f I V ( W ) i s i n f i n i t e .A s s u m e f o r t h e m o m e n t t h a t D h a s f i n it e a r e a . W e s h o w t h a t i f E is c h o s e ns u f f ic i e n t ly s m a l l , t h e n P v i s d i s j o i n t f r o m F f o r e v e r y v ~ I V ( W ) . L e t C b e s o m ec o m p o n e n t o f IV (W ) . L e t j b e s u f f ic i e n tl y l a r g e s o t h a t C i n t e r s e c t s V j , a n d l e t C /b e an y c o m p o n e n t o f C n V j . S i n c e e v e r y c o m p o n e n t o f I 2 ( W ) is i n fi n i te , C i si n f i n i t e , a n d t h e r e f o r e C / m u s t c o n t a i n b o u n d a r y v e r ti c e s o f T j . L e t H j b e t h ec o m p o n e n t o f V ( K ) n V j t h a t c o n t a i n s v 0 . T h e a b o v e a r g u m e n t t e ll s u s t h a t H jc o n t a i n s b o u n d a r y v e r t i c e s o f T j. L e t F * j d e n o t e t h e f a m i l y o f al l s u b s e t s o f V / t h a ti n t e rs e c t e v e r y p a t h i n F 6 j ( H J , C J ). P r o p o s i t i o n 5 .2 n o w i m p l i e s t h a t

E L ( I ' * J ) < V E L 6 j ( H J , C ~ ) - 1 < V E L c ( I V ( K ) , I V (W ) ) - x < E . ( 7 . 3)C o n s i d e r t h e v - m e t r i c m = m j t h a t a s s i g n s t o e a c h v ~ V j t h e d i a m e t e r o f P v .

B y t h e r - fa t n e s s o f t h e s e t s Q v , w e h a v e

a r e a ( P ] ) > r z r m ( v ) 2.T h i s i m p l i e s

a r e a ( m ) < r - l a r - 1 a r e a ( D ) < o0.I n e q u a l i t y (7 . 3 ) n o w i m p l i e s t h a t t h e r e i s s o m e y * ~ F * j s u c h t h a t

L , ~ ( y * ) < ~ / 8 a re a(D )T ar

W e n o w c h o o s e e t o b e s u f f i c ie n t ly s m a l l s o t h a t t h e r i g h t - h a n d s i d e o f t h e a b o v ei n e q u a l it y i s s m a l l e r t h a n d ( F ' , a D ) / 2 , h a l f t h e d i st a n c e f r o m F ' t o d D . S o w e h a v e

d ( F ' , a D )L ' n ( Y * ) < 2 ( 7 . 4 )


18/27

140 Zheng-Xu He and O. SchrammS i n c e t h e s e t s C j a n d H j a r e c o n n e c t e d , L e m m a 4 . 4 i m p li e s th a t t h e r e is a

y ~ ' ~ F * j t h a t i s c o n n e c t e d a n d i s c o n t a i n e d i n y * . L e t Y J = U ~ ~ r r P / . B e c a u s e Wis c o n n e c t e d , w e m a y e s t i m a t e i ts d i a m e t e r a s fo l lo w s :

d i a m e t e r ( W ) < Y '. d i a m e t e r ( P ~ ) = L m ( T * ) d ( F ' , & D ) /2 .S i n ce d i a m e t e r ( Y j ) < d ( F ' , O D ) / 2 < d ( F ' , O D J ) , a n d Y J i n t e r s e c t s O D j , it i s c l e a rt h a t Y J d o e s n o t i n t e r s e c t F ' . S i n c e y ~ ' s e p a r a t e s H j f r o m C j in G j, i t i s c l e a r t h a tY J U 8 D j s e p a r a t e s O o ~ c i P J f r o m P ~ o " S i n c e YJ u c~DJ d o e s n o t i n t e r se c t F ' ,w h i c h i s c o n n e c t e d a n d i n t e r s e c t s P / 0 , i t f o l lo w s t h a t U o ~ c J P [ i s d i s j o in t f r o m F ' .R e c a l l t h a t C j i s a n y c o m p o n e n t o f C n V j , a n d C i s a n y c o m p o n e n t o f I ? ( W ) .T h e r e f o r e , f o r a n y u E 1 .7 (W ) , i f j i s s u f f ic i e n tl y l a r g e s o t h a t u ~ V / a n d d ( F ' , O D J)> d ( F ' , & D ) / 2, t h e n Po n F ' = O . T a k i n g l i m i t s, it f o l l o w s t h a t Pv i s d i s j o i n t f r o mt h e i n te r i o r o f F ' , w h i c h c o n t a i n s F , a n d s o P~ n F = O .

W e s u m m a r i z e o u r c o n c l u s i o n s a s f o ll o w s. F o r e v e r y a c c u m u l a t i o n p o i n t z o f / 5t h e r e i s a n e i g h b o r h o o d W o f z s u c h t h a t Pv A F = O f o r e v e r y u ~ I-~ (I'V ~). L e tW * b e t h e u n i o n o f a ll W z, o v e r al l a c c u m u l a t i o n p o i n t s z o f / 5 . T h e n W * i s a n o p e ns e t t h a t c o n t a i n s t h e a c c u m u l a t i o n p o i n t s o f / 5 . C o n s e q u e n t l y , V - I ? ( W * ) i s f in it e.S i n c e F n Po = ~ f o r a l l v ~ 1 2( W * ), o n l y f in i t e ly m a n y s e t s i n t h e p a c k i n g Pi n t e r s e c t F . H e n c e P i s l o c a l ly f i ni te i n D .

T h i s c o n c l u d e s t h e p r o o f i n t h e c a s e t h a t D h a s f in i t e a r e a . W h e n D h a s in f in i tea r e a , t h e s a m e p r o o f i s v a l i d w h e n t h e s p h e r i c a l m e t r i c o f (~ i s u s e d i n p l a c e o f th ef i a t m e t r i c o f C . T h e o n l y f a c t t o n o t e i s t h a t t h e r e i s s o m e ~ ' a, w h i c h d e p e n d s o n l yo n z , s u c h t h a t t h e s p h e r i c a l a r e a o f P ~ i s a t l e a s t T 1 t im e s t h e s q u a r e o f th es p h e r ic a l d i a m e t e r o f P [ . T h i s f o ll o w s e a s il y f r o m l _ e m m a 3 .3 . T h u s t h e p r o o f i sc o m p l e t e . [ ]

W e c a n n o w p r o v e :7 .2 . Theorem. A disk triangulation graph is C P parabolic i f f i t is VE L parabolic .A disk triangulation graph is C P hyperbolic i f f i t is V E L hyperbolic .Proo f o f Theorems l . 2 and 7 .2 . T h e s e f o ll o w i m m e d i a t e l y f r o m 3 .1 a n d 7 .1 . [ ]

8 . V E L P a r a b o l i c it y , E E L P a r a b o l i c i ty , a n d R e c u r r e n c eW e h a v e s e e n t h a t t h e V E L t y p e o f a d is k tr i a n g u l a t io n g r a p h i s e q u a l t o i ts C P t y pe ,a n d n o w w e es ta b li sh t h e c o n n e c t i o n b e tw e e n V E L a n d E E L t yp e . T h r o u g hT h e o r e m 2 .6 , t hi s r e la t e s t h e C P t y p e o f g r a p h t o w e l l- s t u d ie d n o t i o n s.


19/27

Hype rbolic and Parabolic Packings 1418 . 1 . T h e o r e m . L e t G = ( V , E ) be a localty f inite graph. I f G is E E L parabolic , then i tis also V E L parabolic. Conversely, if G ha s bounded valence an d is VE L parabolic, thenit is EE L parabolic .Proof. S u p p o s e t h a t M is a n e - m e t r i c o n G . D e f i n e a v - m e t r i c m o n G b y

m ( v ) = m a x { M ( [ v ,u ] ) : [ v , u ] ~ E } .I f y i s a n y t r a n s i e n t p a t h i n G , t h e n a s im p l e d i a g o n a l i z a t i o n a r g u m e n t s h o w s t h a tt h e r e i s a p a th 3 " = ( V l , v 2 , . . - ) w i th d i s t i n c t v e r t i c e s t h a t a r e a ll in 3 '. Th u s

oo

L m ( Y ) > > _ L m ( 3 ' ' ) = Y '. m(v y ) > Y'~ M ([ v j , v j + l] ) = Z M ( ' y ' ) .j = l j = l ( 8 . 1 )

F o r e a c h v ~ V l e t e ( v ) d e n o t e a n e d g e e o f G c o n t a i n i n g v t h a t m a x i m i z e sM ( e ) a m o n g s u c h e d g e s . C l e a r ly , e a c h e ~ E i s e q u a l t o e ( v ) f o r a t m o s t t w ov e r t i c e s v . U s in g t h i s , w e g e t

a r e a ( m ) = Y '. m ( v ) 2 = ~ M ( e ( v ) ) zv ~ V v E V

< 2 Y'~ M ( e ) 2 = 2 a r e a ( M ) . ( 8 . 2 )e ~ E

T o g e t h e r w i t h ( 8 .1 ) t h i s es t a b li s h e s t h a t a n E E L p a r a b o l i c g r a p h i s V E L p a r a b o l i c .T o p r o v e t h e o p p o s i t e i m p l i c a t io n , a s s u m e t h a t t h e r e i s a g l o b a l b o u n d k o n t h e

v a le n c e o f a n y v e r te x v ~ V . l ~ t m b e s o m e v - m e t r ic o n G . D e f i n e a n e - m e t r i c Mb y ~ / ( [ u , v ] ) = m a x ( m ( u ) , re(v)) . I t is e a s y to e s t ab l i sh t h a t f o r a n y p a t h y w e h a v eL M ( y ) > L m ( Y ) . M o r e o v e r , s i n c e e a c h v e r t e x is i n c i d e n t w i t h a t m o s t k e d g e s , ac a l c u l a ti o n s i m i l a r t o ( 8 .2 ) g i v e s a r e a ( M ) < k a r e a ( m ) . T h e s e i n e q u a l i ti e s s h o w t h a ta b o u n d e d v a l e n c e V E L p a r a b o l i c g r a p h i s E E L p a r a b o l ic , a n d t h e p r o o f o f t h et h e o r e m is c o m p l e t e . [ ]

8 .2 . Th e o r e m . There is a disk tr iangulation graph which is C P and VE L parabolic , butE E L hyperbolic and transient.

T h i s sh o w s t h a t t h e b o u n d e d v a l e n c e r e q u i r e m e n t i n t h e s e c o n d p a r t o f T h e o r e m8.1 is essentia l .P r o o f . L e t T b e a t r i a n g u l a t i o n o f a n o p e n d is k . I t is n o t h a r d t o s e e t h a t b y a d d i n gv e r t ic e s a n d e d g e s i n si d e t h e t r i a n g u l a r f a c e s o f T a n e w t ri a n g u l a t i o n T * w h o s e1 - s ke l et o n G * i s t r a n s ie n t c a n b e o b t a i n e d . O n t h e o t h e r h a n d , G * is V E L p a r a b o l i ci ff t h e 1 - s k e l e t o n o f T i s V E L p a r a b o l i c . T h e d e t a il s a r e l e f t t o t h e r e a d e r . [ ]

Proof (o f 1 .1) . F o l l o w s i m m e d i a t e l y f r o m T h e o r e m s 8 .1 a n d 7 .2 . [ ]


20/27

142 . Zheng-X u He and O. Sch ram m

9 . P e r i m e t r i c I n e q u a l i t i e s a n d t h e T y p e

9 . 1 . T h e o r e m . L e t G = ( V , E ) b e a l oc a l ly f i n i te , i n fi n it e , c o n n e c t e d g r a p h , l e t W o b ea f i n it e n o n e m p t y s e t o f v e r ti c es o f G , a n d l e t g : [0 , oo) -~ (0 , ~) b e s o m e n o n d e c r e a si n gf u n c t i o n .

(1 ) I f G i s V E L p a r a b o l i c a n d s a t is f ie s t h e p e r i m e t r i c i n e q u a li t yl a W I > g ( l W I ) ( 9 .1 )

f o r e v e r y f i n i t e c o n n e c t e d v e r t e x s e t W D W o, t h e n| 1

] ~ - - = oo ( 9 . 2 )n = 1 g( n) 2 "H e r e a W d e n o t e s t h e s e t o f v e r ti ce s th a t a r e n o t i n W b u t n e i gh b o r w i th s o m ev e r t e x i n I V , a n d [A [ d e n o t e s t h e c a r d i n a l it y o f a s e t A .

(2 ) I f (9 .2 ) h o l d s , a n dlO W kl


21/27

Hype rbol ic and Parabol ic Packings 143I t t u r n s o u t t h a t n ( h ) i s n o t c o n v e n i e n t t o w o r k w i t h , s i n c e i t i s n o t s m o o t h

e n o u g h . W e t h e r e f o r e d e f i n el e n g t h ( I v n [ 0, h ] )s v ( h ) = m ( v ) f o r v ~ V ,

s ( h ) = ~ s v ( h ) .v ~ V

N o t e t h a t s v ( h ) i s e q u a l t o 0 f o r h < r a i n I v , s v ( h ) = 1 f o r h > m a x I v , a n d s ~ i sl i n e a r i n I v . S i n c e d , , ( W o , oo) = 0% i t fo l l o w s t ha t fo r ev e r y h ~ [0, o0) t h e r e a r ef i n i t e l y m a n y v s u c h t h a t I v i n t e r s e c t s [0 , h i . T h e r e f o r e s ( h ) i s a p i e c e w i s e l i n e a rf u n c t i o n . I t s h o u l d b e t h o u g h t o f a s a s m o o t h e d v e r s i o n o f n ( h ) .N o w s e t x )f ( x ) = m in g , -~ . (9 .5 )L e t h ~ [ 0 , ~ ) . I f I m h l >_ s ( h ) / 2 , t h e n

rVhl >_ f ( s ( h ) ) . ( 9 . 6 )S u p p o s e t h a t PV hl < s ( h ) / 2 . T h e n w e h a v e n ( h ) = r Y h f >-- s ( h ) - I g h l > s ( h ) / 2 . C o n -s e q u e n t l y

I g h l > g ( n ( h ) ) > g ( - - - ~ - - ) > f ( s ( h ) ) ,s ( h )

a n d w e c o n c l u d e t h a t ( 9 . 6 ) h o l d s i n a n y c a s e .W e a r e n o w r e a d y to d o s o m e r e a l w o r k . A t p o i n t s h w h e r e s ( h ) i s d i f f e r e n t i a b l e ,

w e h a v ed s 1--d -~ (h ) = E S ' v ( h ) = 2 ~ m ( v ) "

v~V h v~V h

T h e r e f o r e , u s i n g th e C a u c h y - S c h w a r z i n e q u a l i ty (o r t h e i n e q u a li ty b e t w e e n t h ea r i t h m e t i c a n d h a r m o n i c m e a n s ) a n d ( 9 .6 ), w e g e t

d s I V hl 2 f ( s ( h ) ) 2d h - Y ' . { m ( v ) : v ~ Vh } w ( h )

T h i s g i v e sd s d h

- - .f ( s ) 2 >- w ( h )


22/27

144 Zheng-Xu He and O. SchrammI n t e g r a t i n g f o r h i n s o m e i n t e r v a l [ a , b ] , 0 < a < b < oo, a n d u s i n g C a u c h y - S c h w a r za g a i n , w e g e t

f s s(O) ds [ b d h ( b - a ) 2(a) f ($ ) 2 ~ Ja ~ >- f ? w - ( ~ - d h "N o t e t h a t

f w ( h ) d h = fo ~ ., m ( v ) d h = ~ f I m ( v ) d h =v ~ V h v E V vT h e r e f o r e , l e t t i n g b - -> oo in (9 . 7 ) , w e g e t

d s

EvEV

S i n c e( 4 / 4

f ( s ) 2 = m a x 2 , ] g ( s / 2 ) 2 s 2 ,g ( s / 2 ) ~ < - - + - -

( 9 . 7 )

re (v ) 2 = a r e a ( m ) < m.

d m ( W o , O O ) > _ _ d m ( W o , O WN ) >__Ny " g ( n k ) -1 .

k= O

N o t e t h a t

N l a W k l Na r e a ( m ) < E < ) -- , - 1- g ( n k ) 9k= O g ( n k ) 2 k= OS i n ce t h e a b o v e a r e v a l id f o r e a c h N , w e g e t

oo

V E L ( W o , o O ) > ~:~ g ( n k ) -1 .k= O

n k + l - - - - IW k + l l = IW k U ~W kI ~ I W k l + l aW kl ~ n k + g ( n k ) .

( 9 . 8 )

O n t h e o t h e r h a n d ,

W e h a v e

m ( v ) = l g ( n k ) _t f o r v ~ a W k , k < N ,o t h e r w i s e .

w e f i n d t h a t f o g ( s ) - 2 d s = 0% w h i c h i m p l i e s (9 . 2) . T h i s p r o v e s p a r t ( 1 ) .T o e s t a b l i s h p a r t ( 2 ), n o w s e t n k = I W k l , a n d a s s u m e t h a t ( 9 .3 ) h o l d s . L e t N b es o m e p o s i ti v e i n t e g e r , a n d d e f i n e a v - m e t r i c m o n G b y


23/27

Hyperbolic and Parabolic Packings 145

Fig. 9.1. A parabolic graph with exponential growth.Using th i s and the monotonic i ty of g , we obta in

1 1 nk"l--I 1 n k + l - - 1- -> s - -> sg ( n k ) - n k + l - - n k n=nk g ( n ) n ~n k

1 1 nk~l--1 1> Eg ( n k ) g ( n ) - g ( n ) 2"~ : g l k

This impl ies

k=0 n=n0 g ( n ) z "Now part (2) fol lows from (9.8) . [ ]

Th ere i s a cer ta in asym met ry in the two par t s of Th eo re m 9.1 . Whi le the f ir s t par texam ines the re la t ion be tw een the s ize of the bou nda ry of W and the size of W forevery f in ite connec ted ver tex se t W ~ W0, the second par t does th i s only for the se t sW k. This d iffere nce is essent ial ; that is, par t (1) fai ls i f (9.1) is only assu me d for thesets W k. Figure 9.1 gives a disk t r iangulat ion graph, which is essent ial ly equivalent toa graph cons t ru c ted by Soardi [27] , wi th the fo l lowing proper t ies :

(1) G i s VE L parabo l ic .(2) The m axi mu m degre e in G i s 8 .(3 ) IOWkl > C[Wkl for some C > 0 and a l l k = 0 ,1 . . . . . The cons ta nt C does

depe nd on t he cho i ce o f W 0 , bu t no t on k .Proper ty (3) c lear ly impl ies tha t G has exponent ia l growth; i . e . , ]Wk[ >_ (1 + C) k .

10. Deter mining the Type from the ValencesA natura l ques t ion i s, can the type of a graph b e de ter min ed f rom the va lenc es of i tsver t ices . Sup pose fo r a mo me nt tha t G i s a d isk t r i angula t ion graph. I t i s know n [5]tha t i f a ll the ver t ices of G have de gree grea ter than 6 , then G i s not CP parab ol ic ,


24/27

146 Zheng-Xu He and O. Schrammand if all the vertices of G have degree at most 6, then G is CP parabolic. Thefollowing two theorems generalize these results.10.1. Theorem. L e t G b e t h e 1 - s k el e to n o f a n i n f in i t e tr i a n g u la t i o n o f a su r fa c e , a n ds u p p o s e t h a t a t m o s t f i n i t e ly m a n y v e r t ic e s i n G h a v e d e g r e e g r e a t er t h a n 6 . T h e n G isV E L p a r a bo l ic , E E L p a r a b o l ic , a n d r ec u rr e nt .

Due to Th eor em 7.2 this implies that a disk triangul atio n graph with finitely man yvertices of degree greater than 6 is CP parabolic.P r o o f . The met hod of proof is to show that the rate of growth of G is too slow forG to be hyperbolic.

Given a set W c V, we let 0 W den ote the set of v E V - W that ne ighbor withsome vertex in W, and let c~W be the set of vertices in O W that neighbor with somevertex in V - (W u 0W). If K is finite, then OK is a finite set of vertices, and K isdisjoint from the in finite com pone nts of V - OK.

Let K 0 c V be a finite none mpt y set of vertices that conta ins all the vertices in Vof degree gr eater than 6. We define the s eque nce K 1 K 2 .. .. inductively by setting

K n + l = K ~ U O K ~ .Note t hat each K. is finite, and each vertex in 0K n has at least one neighbor in

K n an d at least one neig hbor outside o f Kn 1. Let C. deno te the set of vertices inO K . that have precisely one nei ghbor in K ., and let D~ = O K . - C n . Considersome v ~ 5K .. Let u I . .. .. u m be its neighbors, in circular or der, and set u 0 = u m .Since v has a neighbor in K., we assume without loss of generality that it isu0 = urn. Let j ~ {1 . . . . m - 1} be such that u j q~ K .+ 1 . Since v ~ 5K., such a jexists. Let a be the least index in {1, .. ., j} such that u a ~ K .+ 1 and let b be themaximal index in {j .. . . m - 1} such that u b ~ K~+ 1 Since Ua_ 1 neighbors with vand with u a, and U a_ I ~ K . + 1, it is clear that u a~ D .+ 1 and u ~_ 1 ~ O K . .Similarly, Ub+ 1 ~ O K . and u b ~ D.+ 1 By construction, {u .. .. . u b} contains all theneighbor s of v in aK . + 1.

Suppose for a momen t that v ~ D. n 5 K. . We know that v has at most sixneighbors. Of these, at least two are in K~, and at least two are in O K . , namely,u~_ 1 an d Ub 1. If a ~ b, then v has at least two neighbor s in D.+ 1 namely, u~, u b .As 2 + 2 + 2 --- 6, we see tha t when a :~ b, v has precisely two neig hbor s in D .+ 1and n o nei ghbors in C .+ 1. If a = b, then u~ ---ub is the only neig hbor of v inOKn +1 , and this neighbor is in D. +1. We conclude that a vertex in D. neighborswith at most two vertices in /9. + 1 and with no vertices in C. 1.

The above reasoning also shows that a vertex in C. neighbors with at most threevertices in O K . + 1, of which at most one is in C~ + 1. O ne concl usion that we get is

IC~+ 11_ ICnl. (10.1 )Let mn+ 1 denote the num be r of edges betwee n gn+ 1 and Dn+ 1 On the one hand,m~+ 1 > 2IDa+ 11, bec aus e every vertex in D#+ 1 has at least two ne igh bor s in K. + 1-


25/27

Hype rbolic and Parabolic Packings 147O n t h e o t h e r h a n d , t h e o n l y v e r t i c e s in Kn+ 1 t h a t n e i g h b o r w i th D n + 1 a r e inD n U C . , t h e v e r t i c e s i n D n h a v e a t m o s t t w o n e i g h b o r s i n D . + 1 , a n d t h e v e r t i c e s i nC . h a v e a t m o s t t h r e e n e i g h b o r s i n D n + 1 . T h e r e f o r e ,

21D~+11-< r a n + 1 < 2 1 D ~ l - 4 - 31C=1,w h i c h g i v e s

31CnlID~ +l l _< ID , [ + T (10 .2 )

U s i n g i n d u c t i o n a n d i n e q u a l i t i e s ( 1 0 . 1 ) a n d ( 1 0 . 2 ) , w e s e e t h a t3nlC01IC, ,I _< ICol, IO~l _< IDol + - -2

T h e r e f o r e ,l aK . I = IC , , u D~[ ~ IDo l + (2 n + 1 )lCol . ( 1 0 . 3 )

L e t m b e t h e v - m e t r ic o n G d e f in e d b y r e ( v ) = 1 / ( n l o g n ) f o r v ~ OK n , n > 1,a n d r e ( v ) = 0 f o r v ~ U , > 1 O K n . Si n c e O K n i n t e r s e c ts e v e r y t r a n s i e n t p a t h m e e t -i n g K 0 , w e s e e t h a t dm (K o , o~) >_ ~ > 1 1 / ( n l o g n ) = ~ . O n t h e o t h e r h a n d , ( 10 . 3)i m p li e s t h a t a r e a ( m ) < ~ . H e n c e G i s V E L p a r ab o l ic . F r o m T h e o r e m s 8 .1 a n d 2 .6 i tf o l lo w s t h a t G i s E E L p a r a b o l i c a n d r e c u r r e n t . [ ]

L e t G b e a d is k t r i a n g u l a t i o n g r a p h . F o r v ~ V , l e t d e g ( v ) d e n o t e t h e d e g r e e o fv i n G . T h e a v e r a g e v a l e n c e o f a f in i te n o n e m p t y s e t o f v e r t i c e s W i s j u s t

1a v ( W ) = ~ - ~ Y'~ d e g ( v ) .v ~ W

T h e l o w e r a v e ra g e v a le n c e o f G i s d e f i n e d t o b el a v ( G ) = s u p i n f a v ( W ) ;Wo W~Wo

w h e r e W a n d W 0 a r e n o n e m p t y f in i te c o n n e c t e d s e t s o f v e r ti c e s. ( T h e a u t h o r s d o n o tk n o w i f t h is n o t i o n a p p e a r s i n th e l i t e r a tu r e . )10.2 . T h e o r e m . L e t G b e a l oc a ll y f i n it e c o n n e c t e d p l a n a r g r a ph , a n d s u p p o s e t h a tl a y ( G ) > 6 . T h e n G i s V E L h y p e r b o l ic , a n d t h e re f o r e E E L h y p e r b o li c a n d t r a n si e n t.

N o t e t h a t t h e l o w e r a v e r a g e v a le n c e o f th e h e x a g o n a l g r i d is 6.B e a r d o n a n d S t e p h e n s o n [5 ] h a v e s h o w n t h a t i f e v e r y v e r te x o f G h a s d e g r e e a tl e a s t 7 , t h e n G i s n o t C P p a r a b o l i c . T h e a b o v e t h e o r e m i s a g e n e r a l i z a t i o n o f t h i sre su l t .


26/27

148 Zheng-Xu He and O. SchrammProof. I n a n y f i n it e p l a n a r ~ g ra ph G * w i t h v e r t e x s e t V*, t h e a v e r a g e v a l e n c esa t i s f i e s

a v ( V * ) < 6 . ( 1 0 . 4 )T h i s is a w e l l - k n o w n f a c t b u t f o r t h e c o n v e n i e n c e o f th e n o n e x p e r t r e a d e r s , w e g i vet h e p r o o f h e r e . L e t n , e , f b e t h e n u m b e r o f v e r ti c es , e d g e s, a n d f a c es o f th e g r a p h( w h i c h is e m b e d d e d i n t h e p l a n e ) . T h e E u l e r f o r m u l a g i v es n + f = e + 2 , a n d t h ei n e q u a l i t y 3 f < 2 e h o l d s i f f > 1 , s i n ce e v e r y f a c e m u s t h a v e a t l e a st t h r e e e d g e s o ni ts b o u n d a r y , a n d e a c h e d g e i s o n t h e b o u n d a r y o f a t m o s t t w o f ac e s. F r o m t h e s e i tf o l lo w s th a t n > e / 3 ( a c t u a l l y i t i s t h i s i n e q u a l i t y w h i c h w e n e e d l a t er ) . H o w e v e r ,a v ( V * ) = 2e/n, s i n c e e v e r y e d g e i s c o u n t e d e x a c t ly t w i c e in t h e s u m E v ~ v . d e g ( v ) .T h i s e st a b l i s h e s a v ( V * ) < 6 .

W e n o w r e t u r n t o th e i n f in i te g r a p h G . L e t W 0 b e a f i n i t e c o n n e c t e d n o n e m p t ys e t o f v e r t i c e s s u c h t h a t a v ( W ) > C > 6 f o r s o m e c o n s t a n t C a n d e v e r y f i n it ec o n n e c t e d s e t o f v e r t ic e s W D W 0 . C o n s i d e r s u c h a W , a n d l e t G * b e t h e r e s t r i c t i o no f G t o W U d W ; t h a t is, t h e v e r ti c e s o f G * a r e W u d W , a n d a n e d g e o f G a p p e a r si n G * i f f b o t h i ts e n d p o i n t s a r e i n W U d W . D e n o t e b y n a n d e t h e n u m b e r o fv e r t i c e s a n d e d g e s i n G * , re s p e c t iv e l y . T h e n , d e a r l y , 2 e > I W I a v ( W ) , a n d t h e r e f o r e ,b y t h e p r e v i o u s p a r a g r a p h ,

T h i s g i v e sI W I + I OW I = I W uaW t = n> - 3 >- 6 > IWI.

1 0W I > g ( I W I )w i t h g(x) = (C - 6 ) x / 6 . N o w , s in c e L ':~ =1 g ( n ) - 2 < 0% p a r t ( 1 ) o f T h e o r e m 9 .1s h o w s t h a t G m u s t b e V E L h y p e r b o li c , a n d t h e p r o o f i s c o m p l e t e . [ ]

I t w o u l d b e i n t e r e st i n g t o n a r r o w t h e w i d e g a p b e t w e e n T h e o r e m s 1 0.1 a n d 1 0.2 .S u p p o s e , f o r ex a m p l e , t h a t G is a b o u n d e d v a l e n c e d i sk tr i a n g u l a ti o n g r a p h a n d t h a tv 0 i s s o m e v e r t e x i n G . L e t k n = E v ( 6 - d e g ( v ) ) , w h e r e t h e s u m e x t e n d s o v e r a l lv e r t i c e s v a t d i s t a n c e a t m o s t n f r o m v 0 . C a n c r i t e r

zheng-xu he and o. schramm- hyperbolic and parabolic packings

Documents